Evolutive Algorithm to Optimize the Power Flow in a Network Using Series Compensators-Article

8/8/2019 Evolutive Algorithm to Optimize the Power Flow in a Network Using Series Compensators-Article

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Vr Ve

Z2

Z5

Z1

Z4Z3

Zc

ZeZ

r

Node 1Node 2

Node 3

Line 1

Line 2

Line 5

Line 3 Line 4

Fig. 1. Electric system.

Table I shows the powers (nominal and actual) flowing

across the lines in a normal situation, that is without a line fall.

Table II represents the powers (nominal and actual) flowing

across the lines when line 5 falls, so the problem of this

network is that the line 1 is overloaded (its rated power is 42

MVA and the power flow that crosses it, is 55.958 MVA)whilethe other lines are below their nominal power values, so

the objetive of the algorithms is to modify the power flow

distribution in this network using SSSC converters to ensure

that all lines are below their load limit.

TABLE ITABLE OF POWERS WITHOUT LINE5 FALL.

Line Snominal Sreal1 42 MVA 41.77 MVA

2 81 MVA 50.86 MVA

3 81 MVA 11.95 MVA

4 81 MVA 49.05 MVA

5 72 MVA 50.86 MVA

TABLE IITABLE OF POWERS.

Line Snominal Sreal1 42 MVA 55.958 MVA

2 81 MVA 71.452 MVA

3 81 MVA 6.3349 MVA

4 81 MVA 59.309 MVA

III. PROPOSED EVOLUTIVE ALGORITM.

Fig. 2 shows the structure of the proposed algorithm:

One of the most important things in these types of al-

gorithms is the codification of chromosomes, the proposed

algorithm uses the codification shown in Table III, where Xiare the equivalent impedance of the SSSC converter in the line

i, i the standard deviation of a normal distribution which

determines the mutation of the impedance in the line i. This

represents the structure of a chromosome and every individual

is represented in this way.

Start

Generate

initial

chromosomes

Power flow:

Evaluation

Generar

descendants:

Crossover

Mutate

descendants

Power flow:

Evaluation

Selection

Function

Algorithm

ends?End.

Network

parameters

YesNO

Fig. 2. Structure of the algorithm.

TABLE IIITABLE CHROMOSOMES CODIFICATION.

X1 X2 X3 X4

1 2 3 4

With the lines impedances and the increase or decrease of

this impedance produced by the converter, the other paremeters

to solve the problem: Si, Sci, STC can be calculated.

A. Generate initial chromosomes

At first, the initial individuals must be generated. The

proposed algorithm is an ( + ) evolutive algorithm, with < , that is because in this techniques is recommended to


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use more generated population than initial population. In this

example and are small numbers too, because this problem

is a small one (the network has only three nodes), so it is not

necessary to use a big number of individuals.

The initial choromosomes paremeters: Xi and i aregenerated randomly, this is for all the initial population.

B. Solving Power Flow: Evaluation of the solutions

Now, the other chromosomes parameters Table IV must

be calcultated. With the value of the increased or decreased

impedance in each line, a power flow is solved to obtain the

power in each line, the power of the converters and the total

power installed.

TABLE IVTABLE OF THE NETWORK PAREMETERS TO SOLVE THE PROBLEM.

S1 S2 S3 S4

Sc1 Sc2 Sc3 Sc4

With the power flow, the voltages on the nodes: VN1 andVN2, and the line currents: i1, i2, i3 and i4 can be calculated.

So finally, the powers of the lines are calculated as follows

Eq. 1, Eq. 2, Eq. 3, Eq. 4:

S1 = |VNE I

1| (1)

S2 = |VNE I

2| (2)

S3 = |VNR I

3| (3)

S4 = |VNE I

4 | (4)

To complete the parameters of the network, the power of

the SSSC converters and the total installed power should be

calculated Eq. 5, Eq. 6, Eq. 7, Eq. 8, Eq. 9:

SC1 = |VC1 I

1| (5)

SC2 = |VC2 I

2 | (6)

SC3 = |VC3 I

3 | (7)

SC4 = |VC4 I

4 | (8)

SCTot =

SCi (9)

Finally this calculus is done with all the initial population

to obtain all the parameters of the initial space. To evaluate

wich solution is the best, a fitness function assingns a cost

to each chromosome.

cost =

Sci + (10)

The Eq. 10 shows the form of the fitness function which

has two terms: the total installed power, and a term whichrepresents the cost assigned to each chromosome to overcome

the nominal power value of the lines, called as: penalization

function.

In another point of the algorithm another power flow equal

to this one is solved by the new individuals asigning their

cost too.

C. Generate descendants: Crossover operation

Now the chromosomes descendants should be gener-

ated from crossings between the chromosomes parents.

First, two parents are selected randomly, these chromosomes

generate two descendants; to create the first column of the

Descendant 1 randomly the first column of the Parent 1

or for the Parent 2 was selected (randomly). The first column

of the Descendant 2 is the first column of the other parent(the no selected).

This is done with all the columns and until all the chro-

mosomes descendants are created. The Eq. 3 shows an

example of de generation of two descendants by the crossover

operation.

Fig. 3. Crossing operator.

D. Mutation of the descendants

To solve the problem, its necessary that the impedance

associated with the converters varies, so it is value must

change. In this section it will be explained how this variation

is done. First, two random variables z0 and zi with a normal

distribution with average zero and typical desviation 0 and irespectively, are defined as shown in Eq. 11. The bibliography

recommends the values 0=0.1 and i=0.3.

z0 = 0 N(0, 1)

zi = i N(0, 1) (11)

z0 will remain fixed while zi varies with the expression

Eq. 12 where N(0,1) is a normal distribution with average zero

and typical deviation one:

zin+1 = zin + N(0, 1) (12)

After this, the values of the cumulative probability of the

variation of the impedances: Eq.13 and the new impedances

values X (Eq. 14) are calculated.

in+1 = in exp(z0 + zin+1). (13)

Xin+1 = Xin + in+1 N(0, 1). (14)

This calculation is done for all the values of each chromo-

some impedances, and for all the chromosomes obtained by

the crossing operation.


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IV. SELECTION FUNCTION

In this point, the new initial population should be selected

(between the + chromosomes). The fitness function wasasigned a cost to all the individuals so the chromosomes with

the lower cost are selected to create the new initial population.

The cost of each solution ((Eq.10)) is calculated with two

terms: the penalization funcion asigns a very big cost if

the difference between the nominal powers in the lines andthe powers that flow across the lines is big. This does not

mean that chromosomes that do not comply the power criterion

can not be selected, if there are less than individuals that

comply this criterion, the initial population is completed with

chromosomes that do not meet it, but whose error is the

smallest of it.

As the number of iterations grows, more chromosomes

satisfy the criterion of powers, then the chromosomes that

cause the minimum installed converters power, will be selected

to create the new initial population, that is the first term of the

fitness function.

V. FINALIZATION OF THE ALGORITHM

To end the algorithm, it is necessary to fulfil two conditions.

First, more than chromosomes meets that with their injected

power, all the powers in the lines are below their nominal

value. And second one, the difference between the total

injected power of the best chromosome (this is the one whose

installed power is lower), the average from all the installed

powers of the chromosomes and the the total injected power of

the worse chromosome (this is the one whose installed power

is greater), is less than a certain value Eq.15 and (Eq.16).

STBestParent

1

nSTi

(15)

1

n

STi STWhorseParent

(16)

Using this approach ensures that if the algorithm is ending,

is because it has found an overall minimum, and not a

local one. At the first iterations the difference between those

paremeters is very big, but as the algorithm is doing iterations

that difference is decreasing.

V I. EXPERIMENTAL RESULTS

This algorithm has been programmed in Matlab and its

results have been validated with PsCad. After the algorithm isexecuted, the results of the best chormosome are represented

in Table V.

TABLE VBEST CHROMOSOME.

Xi() 0.0274 1.2156e-15 -3.6313e-13 -2.9358e-15i 2.6518e-14 1.9565e-14 6.2943e-12 1.1020e-14

The parameters of this chromosome indicate that only one

converter in the line 1 is necessary to solve the problem,

because only the impedance of the line 1 has a significant

value. With these values the solution of the power flow of the

network is represented in Table VI.

TABLE VIOTHER PARAMETERS RESULTS.

SCi(V A) 4.3380e5 6.9573e-8 2.6397e-7 -1.0792e-7

SLi(V A) 4.2e7 7.9 801e7 7.987 5e6 6. 3955e7

And the total installed power, that is, the total power of

the converters: SCT=4.3380e5 VA. In this case to solve the

problem it is necesary a converter of aproximately 434 kVA

per phase placed in the line 1.

To see the convergence of the algorithm is possible to see

the evolution of the total installed power in the line 1 Fig.4, in

this figure are represented the solutions obtained by the best

chromosome, the worse chromosome and the mean value of

all the chromosomes.

Fig. 4. Line 1 converter installed power for the first iterations.

Another possible figure is the power flow across line 1,

Fig.5 represents this power for the best chromosome, the worse

chromosome and the mean value of all the chromosomes.

And finally, in Fig.6 it is represented the fitness function

value for the best chromosome and the mean value of all the

chromosomes, so it is posible to see the algorithm evolution.

VII. CONCLUSIONS

This paper presents an evolutive algorithm for choosing the

best placement for a series compensator in order to redistribute

power flow under N-1 contigency. When line 5 falls down the

algorithm recommends to place a series compensator in the

line 1 in order to get a new power flow with all the lines

under their thermal limits.

Besides, this algorithm finds a solution with a small number

of iterations and with good convergence. The next step in de-

veloping is the application of this algorithm to more complex

electric grid systems.


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Fig. 5. Power flow across the line 1.

Fig. 6. Evolution of the algorithm: fitness function.

ACKNOWLEDGEMENT

This work was supported by the Spanish Ministry of

Education and Science under Project ENE2006-02930. The

authors want to thank Red Electrica de Espana for the technical

support.

REFERENCES

[1] G. Breuer, Flexible ac transmission systems: Technology for thefuture, in Proceedings of 20th Annual Electrical / Electronics InsulationConference, Boston,, October 7-10, 1991.

[2] N. Hingorani, Flexible ac transmission, IEEE Spectrum, vol. 30, pp.4045, 1993.

[3] O. Anaya-Lara and E. Acha, Modeling andanalysis of custom powersystems by pscad/emtdc, IEEE Transactions, Power Delivery, vol. 17,pp. 266272, 2002.

[4] L. Gyugyi, C. D. Schauder, and K. K. Sen, Static series compensator:a solid-state approach to the series compensation of transmission lines,

IEEE Transactions on Power Delivery, vol. 12, pp. 406407, 1997.

[5] M. Noroozian and G. Andersson, Power flow control by use ofcontrollable series components, IEEE Transactions on Power Delivery,vol. 8, pp. 1218, 1993.

[6] C.-C. L. Ying-Yi Hong, A heuristic and algorithmic approach to varplanning, IEEE Trans. on Power Systems, vol. Vol. 7, No. 2, p. 505512, 1992.

[7] H.-D. C. Y.-L. C. Ying-Tung Hsiao, Chung-Chang Liu, A new approachfor optimal var sources planning in large scale electric power systems,

IEEE Trans. on Power Systems, vol. Vol. 8, No. 3, p. 988 996, 1993.[8] C.-l. H. Yann-Chan Huang, Hong-Tzer Yang, Solving the capacitor

placement problem in a radial distribution system using tabu searchapproach, IEEE Trans. on Power Systems, vol. Vol. 11, No. 4, p. 1868

1873, 1996.[9] J. T. M. L. L. Lai, Application of evolutionary programming to reactivepower planning-comparison with nonlinear programming approach,

IEEE Trans. on Power Systems,, vol. Vol. 12, No. 1, p. 198206, 1997.[10] Y.-M. P. K.Y. Lee, B. Xiaomin, Optimization method for reactive power

planning by using a modified simple genetic algorithm, IEEE Trans.on Power Systems, vol. Vol. 10, No. 4, p. 1843 1850, 1995.

[11] H.-D. C. Y.-L. C. Ying-Tung Hsiao, Chung-Chang Liu, A new approachfor optimal var sources planning in large scale electric power systems,

IEEE Trans. on Power Systems, vol. Vol. 8, No. 3, p. 988 996, 1993.[12] P. M. M. M. M. Delfanti, G.P. Granelli, Optimal capacitor placement

using deterministic and genetic algorithms, IEEE Trans. on PowerSystems, vol. Vol. 15, No 3, p. 10411046, 2000.

[13] Y.-L. Chen, An interactive fuzzy-norm satisfying method for multiob-jective reactive power sources planning, IEEE Trans. on Power Systems,vol. Vol. 15, No. 3, pp. 11541150, 2000.

[14] H. A. F. Benemar Alencar de Souza, Helton do Nascimento alves, Mi-

crogenetic algorithms and fuzzy logic applied to the optimal placementof capacitor banks in distribution networks, IEEE Trans. on PowerSystems, vol. Vol. 19, No. 2, p. 942 947, 2004.

[15] P. A. Sundhararajan, Srinivasan, Optimal selection of capacitors forradial distribution systems using a genetic algorithm, IEEE Trans. onPower Systems, vol. Vol. 9, No 3, p. 1499 1507, 1994.

[16] T. S. C. H. C. Leung, Optimal placement of facts controller in powersystem by a genetic-based algorithm, in International Conference onPower Electronics and Drive Systems, PEDS99, 1999.

[17] F. Y. K.Y. Lee, Optimal reactive power planning using evolutionaryalgorithms: a comparative study for evolutionary programming, evo-lutionary strategy, genetic algorithm, and linear programming, IEEETrans. on Power Systems, vol. Vol. 13, No. 1, pp. 101108, 1998.

[18] H. M. C. J. L. G. M. L. P. M. A. Diaz, F. Glover and F. T. Tseng,Optimizacin heurstica y redes neuronales en direccin de operaciones e

ingeniera. Editorial Parainfo, 1996.

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Evolutive Algorithm to Optimize the Power Flow in a Network Using Series Compensators-Article